In a dead ball-dominated contest, Gary O'Neil's troops deservedly went ahead through Pablo Sarabia's penalty, but the Irons improved after the break and restored parity through Lucas Paqueta's own spot kick.
A freak James Ward-Prowse goal direct from a corner then completed the visitors' comeback, although Max Kilman thought he had stolen a point for Wolves right at the end, only for the officials to disallow his effort for offside, sparking furious scenes.
Despite Matheus Cunha's return to full fitness, Wolves head coach O'Neil went with an incredibly defensive setup boasting no recognised striker, while Edson Alvarez unsurprisingly returned to the West Ham XI straight away following his suspension, displacing Michail Antonio.
Rayan Ait-Nouri - scorer of three goals in his last four games - was one of the most advanced players for the hosts and had an early crack with just three minutes gone, but his effort was too high to trouble Lukasz Fabianski.
West Ham were content to sit deep, but the Hammers should have made more of a promising opening in the 12th minute, when Jarrod Bowen robbed Nelson Semedo of possession and cut back for Tomas Soucek, but the Czech midfielder got his feet caught in a tangle and the chance went begging.
Wolves soon began huffing and puffing again, but Fabianski was proving unbeatable, coming out quickly to thwart Sarabia in the 20th minute before comfortably denying Tommy Doyle from a tight angle three moments later.
The hosts' endeavours were eventually rewarded in the 33rd minute from the penalty spot, though, as Emerson Palmieri wiped out fellow full-back Ait-Nouri, and Sarabia just about managed to squeeze his spot kick past Fabianski via the inside of the post.
The Spaniard's conversion came from the 10th penalty that West Ham had conceded in the current Premier League season, and the sub-par Irons could not muster up a response in the short time that remained during the first half.
Alterations on the visitors' end therefore came as no shock - Antonio and Ben Johnson were introduced - and the former had West Ham's first shot on target just one minute into the second period, although it was meat and drink for Jose Sa.
Injuries to key men then reared their ugly heads for both sides, as both Bowen and Ait-Nouri were forced off within two minutes of each other, although West Ham displayed a marked improvement from their lukewarm first-half display.
David Moyes's men were rewarded in the 63rd minute when Emerson nodded in from Mohammed Kudus's cross to the back stick, but the officials saw fit to disallow the strike for the ex-Chelsea man's perceived foul on Semedo.
Emerson could count himself unfortunate to see his effort chalked off, but the Italian would finally make amends for his first-half penalty giveaway in the Wolves area, as his cross led to a handball from Kilman.
In a carbon copy of Sarabia's first-half effort, Sa dived the right way, but Paqueta's spot kick was perfectly placed into the bottom right-hand corner, and West Ham's leveller was richly deserved.
With just six minutes of normal time remaining, the resurgent Hammers turned the contest on its head via another dead-ball situation, albeit in far more unorthodox circumstances, as Ward-Prowse's corner sailed right over the outstretched arm of Sa and flew into the far side of the net.
The hosts were granted 10 minutes of additional time to try to claw their way back, the ninth of which saw Kilman nod home from a corner, but young winger Tawanda Chirewa was deemed to have been obstructing Fabianski from an offside position.
Referee Tony Harrington disallowed the goal after a quick check of the pitchside monitor, making himself Molineux public enemy number one in the process, feeling the force of a cacophony of boos and also receiving an earful from Cunha, which the Brazilian picked up a post-game booking for.
Despite losing in excruciating circumstances, Wolves remain 11th thanks to Bournemouth and Fulham's defeats, although they are now six points behind their seventh-placed visitors, who are only behind Manchester United on goal difference having played two games more.
Wolves are back in action at the same time next Saturday away to Nottingham Forest, while West Ham have continental matters to attend to, travelling to Bundesliga champions-elect Bayer Leverkusen on Thursday in the first leg of their quarter-final.